P
US7558089B2ExpiredUtilityPatentIndex 80

Method and apparatus for space vector modulation in multi-level inverters

Assignee: ADVANCED ENERGY CONVERSION LLCPriority: Dec 13, 2005Filed: Dec 13, 2006Granted: Jul 7, 2009
Est. expiryDec 13, 2025(expired)· nominal 20-yr term from priority
Inventors:MESE ERKANTORREY DAVID ALLAN
H02M 7/53876H02M 7/487
80
PatentIndex Score
15
Cited by
20
References
8
Claims

Abstract

An apparatus for space vector modulation of a multi-level inverter, the apparatus comprising: a triangle generator for finding a modulation triangle enclosing a reference voltage vector, the modulation triangle comprising vertex vectors; a duty ratio calculator for determining duty ratios for the vertex vectors; a switching function selector for selecting switching functions corresponding to the vertex vectors; a forbidden transition identifier for identifying a forbidden transition between two of the vertex vectors; and a control module for applying the switching functions to the multi-level inverter according to the duty ratios and avoiding the forbidden transition.

Claims

exact text as granted — not AI-modified
1. An apparatus for space vector modulation of a multi-level inverter, said apparatus comprising:
 a triangle generator for finding a modulation triangle enclosing a reference voltage vector, said modulation triangle comprising vertex vectors; 
 a duty ratio calculator for determining duty ratios for said vertex vectors; 
 a forbidden transition identifier for identifying a forbidden transition between two of said vertex vectors; 
 a control module for applying said switching functions to said multi-level inverter according to said duty ratios and avoiding said forbidden transition; and 
 a switching function selector for selecting switching functions corresponding to said vertex vectors, wherein said switching function selector comprises: 
 a vertex level computer for computing vertex levels corresponding to said vertex vectors, each of said vertex vectors being described by a respective skew coordinate pair, (G, H), each of said vertex levels, L, being computed according to the rule: 
 
     
       
         
           
             
               
                 
                   L 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 abs 
                                 ⁡ 
                                 
                                   ( 
                                   H 
                                   ) 
                                 
                               
                               , 
                               
                                 
                                   if 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   G 
                                 
                                 = 
                                 
                                   
                                     
                                       0 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       or 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       G 
                                     
                                     + 
                                     H 
                                   
                                   = 
                                   0 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 abs 
                                 ⁡ 
                                 
                                   ( 
                                   G 
                                   ) 
                                 
                               
                               , 
                               
                                 
                                   if 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   H 
                                 
                                 = 
                                 0 
                               
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         2 
                                         ⁢ 
                                         
                                           π 
                                           / 
                                           3 
                                         
                                       
                                       ) 
                                     
                                   
                                   / 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       T 
                                       ) 
                                     
                                   
                                 
                               
                               , 
                               otherwise 
                             
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       where 
                     
                   
                 
               
             
             
               
                 
                   T 
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           H 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                   π 
                                   / 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                         
                           G 
                           + 
                           
                             H 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   π 
                                   / 
                                   3 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     and “abs( )” denotes the absolute value function:
 a triangle level computer for computing a triangle level, K, equal to the maximum of said vertex levels; 
 a switching function computer for computing said switching functions, each of said switching functions, V, being computed according to the rule: 
 
     
       
         
           
             V 
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   K 
                                   , 
                                   
                                     abs 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         K 
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             G 
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   , 
                                   
                                     abs 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         K 
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             G 
                                             ) 
                                           
                                         
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             H 
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 G 
                               
                               > 
                               
                                 0 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 and 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     G 
                                     + 
                                     H 
                                   
                                   ) 
                                 
                               
                               > 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 abs 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     K 
                                     - 
                                     
                                       abs 
                                       ⁡ 
                                       
                                         ( 
                                         G 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               , 
                               K 
                               , 
                               
                                 abs 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     K 
                                     - 
                                     
                                       abs 
                                       ⁡ 
                                       
                                         ( 
                                         H 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           , 
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               G 
                             
                             < 
                             
                               0 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               and 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               H 
                             
                             > 
                             0 
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     abs 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         K 
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             G 
                                             ) 
                                           
                                         
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             H 
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   , 
                                   
                                     abs 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         K 
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             H 
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   , 
                                   K 
                                 
                                 ] 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 H 
                               
                               < 
                               
                                 0 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 and 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     G 
                                     + 
                                     H 
                                   
                                   ) 
                                 
                               
                               < 
                               0 
                             
                           
                         
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
   
   
     2. The apparatus of  claim 1  wherein said forbidden transition identifier comprises:
 a forbidden transition module for identifying as forbidden a transition between a first vertex vector, having skew coordinates (G 1 ,H 1 ), and a second vertex vector, having skew coordinates (G 2 ,H 2 ) if
   ( G   1 >0 and  G   1   +H   1 >0 and  G   2 >0 and  G   2   +H   2 >0 and  H   1   −H   2 =0) 
   or 
   ( G   1 <0 and  H   1 >0 and  G   2 <0 and  H   2 >0 and  G   1   −G   2   +H   1   −H   2 =0) 
   or 
   ( H   1 <0 and  G   1   +H   1 <0 and  H   2 <0 and  G   2   +H   2 <0 and  G   1   −G   2 =0). 
 
 
   
   
     3. The apparatus of  claim 1  wherein said control module is configured for performing the acts of:
 denoting said switching functions respectively as V 1 , V 2 , V 3 , said forbidden transition being between V 1  and V 3 ; and 
 repeating said switching functions in the order V 1 , V 2 , V 3 , V 3 , V 2 , V 1 . 
 
   
   
     4. An apparatus for space vector modulation of a multi-level inverter system, said apparatus comprising:
 a triangle generator for finding a modulation triangle enclosing a reference voltage vector, said modulation triangle comprising vertex vectors; 
 a duty ratio calculator for determining duty ratios for said vertex vectors; 
 a switching function selector for selecting switching functions corresponding to said vertex vectors; 
 a forbidden transition identifier for identifying a forbidden transition between two of said vertex vectors; and 
 a control module for applying said switching functions to said multi-level inverter according to said duty ratios and avoiding said forbidden transition, 
 said switching function selector comprising: 
 a vertex level computer for computing vertex levels corresponding to said vertex vectors, each of said vertex vectors being described by a respective skew coordinate pair, (G, H), each of said vertex levels, L, being computed according to the rule: 
 
     
       
         
           
             
               
                 
                   L 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 abs 
                                 ⁡ 
                                 
                                   ( 
                                   H 
                                   ) 
                                 
                               
                               , 
                               
                                 
                                   if 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   G 
                                 
                                 = 
                                 
                                   
                                     
                                       0 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       or 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       G 
                                     
                                     + 
                                     H 
                                   
                                   = 
                                   0 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 abs 
                                 ⁡ 
                                 
                                   ( 
                                   G 
                                   ) 
                                 
                               
                               , 
                               
                                 
                                   if 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   H 
                                 
                                 = 
                                 0 
                               
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         2 
                                         ⁢ 
                                         
                                           π 
                                           / 
                                           3 
                                         
                                       
                                       ) 
                                     
                                   
                                   / 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       T 
                                       ) 
                                     
                                   
                                 
                               
                               , 
                               otherwise 
                             
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       where 
                     
                   
                 
               
             
             
               
                 
                   T 
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           H 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                   π 
                                   / 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                         
                           G 
                           + 
                           
                             H 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   π 
                                   / 
                                   3 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     and “abs( )” denotes the absolute value function;
 a triangle level computer for computing a triangle level, K, equal to the maximum of said vertex levels; 
 a switching function computer for computing said switching functions, each of said switching functions, V, being computed according to the rule: 
 
     
       
         
           
             V 
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             
                               [ 
                               
                                 K 
                                 , 
                                 
                                   abs 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         K 
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             G 
                                             ) 
                                           
                                         
                                       
                                       , 
                                       
                                         abs 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             K 
                                             - 
                                             
                                               abs 
                                               ⁡ 
                                               
                                                 ( 
                                                 G 
                                                 ) 
                                               
                                             
                                             - 
                                             
                                               abs 
                                               ⁡ 
                                               
                                                 ( 
                                                 H 
                                                 ) 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               G 
                             
                             > 
                             
                               0 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               and 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   G 
                                   + 
                                   H 
                                 
                                 ) 
                               
                             
                             > 
                             0 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               abs 
                               ⁡ 
                               
                                 ( 
                                 
                                   K 
                                   - 
                                   
                                     abs 
                                     ⁡ 
                                     
                                       ( 
                                       G 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                             , 
                             K 
                             , 
                             
                               abs 
                               ⁡ 
                               
                                 ( 
                                 
                                   K 
                                   - 
                                   
                                     abs 
                                     ⁡ 
                                     
                                       ( 
                                       H 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         , 
                         
                           
                             if 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             G 
                           
                           < 
                           
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             and 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             H 
                           
                           > 
                           0 
                         
                       
                       ⁢ 
                       
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               [ 
                               
                                 
                                   abs 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       K 
                                       - 
                                       
                                         abs 
                                         ⁡ 
                                         
                                           ( 
                                           G 
                                           ) 
                                         
                                       
                                       - 
                                       
                                         abs 
                                         ⁡ 
                                         
                                           ( 
                                           H 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 , 
                                 
                                   abs 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       K 
                                       - 
                                       
                                         abs 
                                         ⁡ 
                                         
                                           ( 
                                           H 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 , 
                                 K 
                               
                               ] 
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               H 
                             
                             < 
                             
                               0 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               and 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   G 
                                   + 
                                   H 
                                 
                                 ) 
                               
                             
                             < 
                             0 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       said forbidden transition identifier comprising: 
       a forbidden transition module for identifying as forbidden a transition between a first vertex vector, having skew coordinates (G 1 ,H 1 ), and a second vertex vector, having skew coordinates (G 2 ,H 2 ) if
   ( G   1 >0 and  G   1   +H   1 >0 and  G   2 >0 and  G   2   +H   2 >0 and  H   1   −H   2 =0) 
   or 
   ( G   1 <0 and  H   1 >0 and  G   2 <0 and  H   2 >0 and  G   1   −G   2   +H   1   −H   2 =0) 
   or 
   ( H   1 <0 and  G   1   +H   1 <0 and  H   2 <0 and  G   2   +H   2 <0 and  G   1   −G   2 =0). 
 
     
   
   
     5. The apparatus of  claim 4  wherein said control module is configured for performing the acts of:
 denoting said switching functions respectively as V 1 , V 2 , V 3 , said forbidden transition being between V 1  and V 3 ; and 
 repeating said switching functions in the order V 1 , V 2 , V 3 , V 3 , V 2 , V 1 . 
 
   
   
     6. A method of space vector modulation for a multi-level inverter system, said method comprising the acts of:
 finding a modulation triangle enclosing a reference voltage vector, said modulation triangle comprising vertex vectors; 
 determining duty ratios for said vertex vectors; 
 selecting switching functions corresponding to said vertex vectors; 
 identifying a forbidden transition between two of said vertex vectors; 
 applying said switching functions to said multi-level inverter according to said duty ratios and avoiding said forbidden transition; and 
 selecting switching functions corresponding to said vertex vectors, such act comprising: 
 computing vertex levels corresponding to said vertex vectors, each of said vertex vectors being described by a respective skew coordinate pair, (G, H), each of said vertex levels, L, being computed according to the rule: 
 
     
       
         
           
             
               
                 
                   L 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 abs 
                                 ⁡ 
                                 
                                   ( 
                                   H 
                                   ) 
                                 
                               
                               , 
                               
                                 
                                   if 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   G 
                                 
                                 = 
                                 
                                   
                                     
                                       0 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       or 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       G 
                                     
                                     + 
                                     H 
                                   
                                   = 
                                   0 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 abs 
                                 ⁡ 
                                 
                                   ( 
                                   G 
                                   ) 
                                 
                               
                               , 
                               
                                 
                                   if 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   H 
                                 
                                 = 
                                 0 
                               
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         2 
                                         ⁢ 
                                         
                                           π 
                                           / 
                                           3 
                                         
                                       
                                       ) 
                                     
                                   
                                   / 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       T 
                                       ) 
                                     
                                   
                                 
                               
                               , 
                               otherwise 
                             
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       where 
                     
                   
                 
               
             
             
               
                 
                   T 
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           H 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                   π 
                                   / 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                         
                           G 
                           + 
                           
                             H 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   π 
                                   / 
                                   3 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     and “abs( )” denotes the absolute value function;
 computing a triangle level, K, equal to the maximum of said vertex levels; 
 computing said switching functions, each of said switching functions, V, being computed according to the rule: 
 
     
       
         
           
             V 
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   K 
                                   , 
                                   
                                     abs 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         K 
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             G 
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   , 
                                   
                                     abs 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         K 
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             G 
                                             ) 
                                           
                                         
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             H 
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 G 
                               
                               > 
                               
                                 0 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 and 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     G 
                                     + 
                                     H 
                                   
                                   ) 
                                 
                               
                               > 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 abs 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     K 
                                     - 
                                     
                                       abs 
                                       ⁡ 
                                       
                                         ( 
                                         G 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               , 
                               K 
                               , 
                               
                                 abs 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     K 
                                     - 
                                     
                                       abs 
                                       ⁡ 
                                       
                                         ( 
                                         H 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           , 
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               G 
                             
                             < 
                             
                               0 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               and 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               H 
                             
                             > 
                             0 
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     abs 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         K 
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             G 
                                             ) 
                                           
                                         
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             H 
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   , 
                                   
                                     abs 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         K 
                                         - 
                                         
                                           abs 
                                           ⁡ 
                                           
                                             ( 
                                             H 
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   , 
                                   K 
                                 
                                 ] 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 H 
                               
                               < 
                               
                                 0 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 and 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     G 
                                     + 
                                     H 
                                   
                                   ) 
                                 
                               
                               < 
                               0 
                             
                           
                         
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
   
   
     7. The method of  claim 6  wherein said act of identifying a forbidden transition between two of said vertex vectors comprises:
 identifying as forbidden a transition between a first vertex vector, having skew coordinates (G 1 ,H 1 ), and a second vertex vector, having skew coordinates (G 2 ,H 2 ) if
   ( G   1 >0 and  G   1   +H   1 >0 and  G   2 >0 and  G   2   +H   2 >0 and  H   1   −H   2 =0) 
   or 
   ( G   1 <0 and  H   1 >0 and  G   2 <0 and  H   2 >0 and  G   1   −G   2   +H   1   −H   2 =0) 
   or 
   ( H   1 <0 and  G   1   +H   1 <0 and  H   2 <0 and  G   2   +H   2 <0 and  G   1   −G   2 =0). 
 
 
   
   
     8. The method of  claim 6  wherein said act of applying said switching functions to said multi-level inverter comprises:
 denoting said switching functions respectively as V 1 , V 2 , V 3 , said forbidden transition being between V 1  and V 3 ; and 
 repeating said switching functions in the order V 1 , V 2 , V 3 , V 3 , V 2 , V 1 .

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